I worked on a John Deere that has one of the pulleys that pivots to take up the slack when you engage the deck. It has an adjustment to add or remove slack.
This deck shook a lot from loose spindle bearings. Replaced all 4 and new belt. Running like new now!! Some mowers have adjustments that...
It does sound like a solenoid safety ground problem. Ground the terminal back to the negative side of the battery. Should crank!! Or got a defective new solenoid or the heavy battery to solenoid wire is defective.